| L(s) = 1 | + 3·4-s − 2·5-s − 2·7-s − 6·11-s − 6·13-s + 5·16-s + 8·17-s − 2·19-s − 6·20-s − 8·23-s + 3·25-s − 6·28-s − 2·29-s + 12·31-s + 4·35-s + 2·37-s + 14·41-s + 6·43-s − 18·44-s − 8·47-s − 4·49-s − 18·52-s − 12·53-s + 12·55-s − 12·59-s − 12·61-s + 3·64-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s − 0.755·7-s − 1.80·11-s − 1.66·13-s + 5/4·16-s + 1.94·17-s − 0.458·19-s − 1.34·20-s − 1.66·23-s + 3/5·25-s − 1.13·28-s − 0.371·29-s + 2.15·31-s + 0.676·35-s + 0.328·37-s + 2.18·41-s + 0.914·43-s − 2.71·44-s − 1.16·47-s − 4/7·49-s − 2.49·52-s − 1.64·53-s + 1.61·55-s − 1.56·59-s − 1.53·61-s + 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.268930006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.268930006\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 24 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 124 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.67271511362287841677187630475, −10.02581513974420537155598790993, −9.579828267433626812097760759297, −9.516934784813309500688880154964, −8.419923197513881581540426457735, −7.80089205681565877940778521648, −7.80062728705847226041248418507, −7.71036418310382896784177414998, −7.11459669919728983154610827023, −6.29703094192450178260044183745, −6.27725230823798878770474023582, −5.71537394554387100462234331872, −5.05133025326034824774734512449, −4.64581092554730096159220447587, −4.02563579301204193992383587525, −3.08520710724224944364398989554, −2.98971781948011649321538550987, −2.50442802574960914334913958952, −1.79827696998905858967763090168, −0.50982924289492545929628281306,
0.50982924289492545929628281306, 1.79827696998905858967763090168, 2.50442802574960914334913958952, 2.98971781948011649321538550987, 3.08520710724224944364398989554, 4.02563579301204193992383587525, 4.64581092554730096159220447587, 5.05133025326034824774734512449, 5.71537394554387100462234331872, 6.27725230823798878770474023582, 6.29703094192450178260044183745, 7.11459669919728983154610827023, 7.71036418310382896784177414998, 7.80062728705847226041248418507, 7.80089205681565877940778521648, 8.419923197513881581540426457735, 9.516934784813309500688880154964, 9.579828267433626812097760759297, 10.02581513974420537155598790993, 10.67271511362287841677187630475
